The concept of a strong \(a\)-valuation was introduced by Maheo, who showed that if a graph \(G\) has a strong \(a\)-valuation, then so does \(G \times K_2\). We show that for various graphs \(G\), \(G \times Q_n\) has a strong \(a\)-valuation and \(G \times P_n\) has an \(a\)-valuation, where \(Q_n\) is the \(n\)-cube and \(P_n\) the path with \(n\) edges, including \(G = K_{m,2}\) for any \(m\). Yet we show that \(K_{m,n} \times K_2\) does not have a strong \(a\)-valuation if \(m\) and \(n\) are distinct odd integers.